Stability types; EAC for parallel lines

Definitions

• Power system stability: the ability of a power system to stay in a state of operating equilibrium under normal conditions and to regain an acceptable equilibrium after a disturbance, with all synchronous machines remaining in synchronism.
• Steady state stability: ability to remain in synchronism for small, gradual changes in load; limit $P_{max} = EV/X$ at $\delta = 90^\circ$, with stable operation where $dP_e/d\delta > 0$.
• Dynamic stability: ability to remain stable for small disturbances over a longer time (several seconds to minutes) when the effects of control systems (AVR, governor, PSS) are included; it concerns damping of low-frequency oscillations.
• Transient stability: ability to remain in synchronism after a large, sudden disturbance (fault, loss of a line, loss of a generator); judged over the first swing (about 1 s).
• Voltage stability: ability of the system to maintain acceptable voltages at all buses after a disturbance; it is lost when reactive power demand cannot be met, leading to voltage collapse.

Equal area criterion for parallel lines

System: generator ($E$, $X_d'$) feeding an infinite bus ($V$) through two parallel lines of reactance $X_L$ each. A 3-phase fault occurs on one line and is cleared by opening the breakers at both ends of that line.

          +---[CB]--- X_L ---[CB]---+
 E--Xd'--(1)                       (2)--- V (inf. bus)
          +---[CB]--- X_L -x-[CB]---+
                           F

Three power-angle curves:

$$\begin{aligned} \text{Pre-fault: } & P_{e1} = \frac{EV}{X_d' + X_L/2}\sin\delta = P_{max1}\sin\delta \\ \text{During fault: } & P_{e2} = \frac{EV}{X_{tr}}\sin\delta = P_{max2}\sin\delta \quad (X_{tr} \text{ from star-delta reduction; } P_{max2}=0 \text{ if F is at bus 1 or 2}) \\ \text{Post-fault: } & P_{e3} = \frac{EV}{X_d' + X_L}\sin\delta = P_{max3}\sin\delta \end{aligned}$$

with $P_{max1} > P_{max3} > P_{max2}$.

Procedure:

1. Initial angle: $\delta_0 = \sin^{-1}(P_m/P_{max1})$.
2. During the fault, $P_m > P_{e2}$; the rotor accelerates from $\delta_0$ to the clearing angle $\delta_c$. Accelerating area: $A_1 = \int_{\delta_0}^{\delta_c}(P_m - P_{max2}\sin\delta)\,d\delta$
3. After clearing, the operating point jumps to the post-fault curve; $P_{e3} > P_m$, the rotor decelerates. Decelerating area: $A_2 = \int_{\delta_c}^{\delta_1}(P_{max3}\sin\delta - P_m)\,d\delta$
4. The system is stable if $A_2 = A_1$ is reached at some $\delta_1 < \delta_{max} = \pi - \sin^{-1}(P_m/P_{max3})$.
5. Setting $A_1 = A_2$ with $\delta_1 = \delta_{max}$ gives the critical clearing angle:

$$\cos\delta_{cr} = \frac{P_m(\delta_{max}-\delta_0) + P_{max3}\cos\delta_{max} - P_{max2}\cos\delta_0}{P_{max3} - P_{max2}}$$

 Pe
  ^          Pmax1 sin d (pre-fault)
  |           .-''-.
  |         .' .--. '.    Pmax3 sin d
  |        / .' A2 '. \   (post-fault)
 Pm-------+-+-------+--\------
  |      /|A1|        \ \
  |     / |..|.....     \ \  Pmax2 sin d
  +----+--+--+--------+--+--> d
       0  d0 dc     dmax 180
 A1: accelerating area, A2: decelerating area

Having two parallel lines is itself a stability aid: after one line trips, the other still carries power ($P_{max3} > 0$), so a decelerating area exists.